chapter 22 collective dynamics of pedestrians with no ...€¦ · chapter 22 collective dynamics of...

Chapter 22Collective Dynamics of Pedestrians with NoFixed Destination

Takayuki Hiraoka, Takashi Shimada, and Nobuyasu Ito

Abstract In order to understand pedestrian dynamics, we construct a modelof self-propelled disk particles interacting repulsively with no fixed destination.From molecular dynamics simulations, we find that the model exhibits collectivemotion and transition from a disordered to a polar-ordered, heterogenous state.Binary scattering study suggests that ordering originates from parallel alignmentof particles’ velocity after collision. The dependency of alignment tendency on themodel parameter agrees well with the behavior of multiparticle systems. We verifythat the model reproduces the actual pedestrian phenomena in a straight pathway.Although there is still a gap with empirical findings, especially in high densities, theresult implies that pedestrian crowds can spontaneously build up a collective motioneven in the situation where they have lost their destinations.

22.1 Introduction

From schooling of fish, flocking of birds, swarming of insects to migration ofcells or bacteria, we often observe collective behaviors of biological organisms.One may presume that these fascinating pattern formation in nature is producedby sophisticated information processing mechanism specific to the species, byelaborate interaction between individuals, or by presence of a special individual who

T. Hiraoka (�) • T. ShimadaDepartment of Applied Physics, Graduate School of Engineering, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

JST CREST, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japane-mail: [email protected]; [email protected]

N. ItoDepartment of Applied Physics, Graduate School of Engineering, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

JST CREST, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan

RIKEN AICS, 7-1-26, Minatojima-minami-machi, Chuo-ku, Kobe, Hyogo 650-0047, Japane-mail: [email protected]

© The Author(s) 2015H. Takayasu et al. (eds.), Proceedings of the International Conference on SocialModeling and Simulation, plus Econophysics Colloquium 2014, SpringerProceedings in Complexity, DOI 10.1007/978-3-319-20591-5_22

243

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244 T. Hiraoka et al.

takes leadership. However, recent studies on self-propelled particle systems revealedthat collective motion could arise even in the absence of long-range communication,complex behavioral rules, and global leadership [1–3]. Furthermore, it has beenshown that the non-equilibrium character enables the systems to develop long-rangeorder and anomalously large density fluctuations, which are unusual in equilibriumsystems [4–8]. Recent studies have found these features are shared not only bybiological organisms but also by non-biological systems such as vibrated granularparticles, in which explicit alignment with neighbors are absent [9].

One of the important research subject studied under the concept of the self-propelled particle is the pedestrian dynamics. We notice that there are characteristicpatterns of crowds in streets, intersections, train stations, airport terminals, concerthalls, sport stadiums, political demonstrations, etc. Such patterns in the urban envi-ronment spontaneously arise from individuals moving with their own destinationsor intentions. Many microscopic models have been proposed to describe pedestrianmovement. They can be categorized into two main types: cellular automata models[10, 11], in which time and space is discretized, have an advantage in computationalcost, while force-based models, which is inspired by Newtonian mechanics, cansimulate realistic trajectories [12, 13]. Excluded volume interaction and repulsiondue to social psychological effect play an important role in both types of model.

In this proceedings we aim to establish a kinetic understanding towards thecollective dynamics of self-propelled particle systems with repulsive interaction.In order to find out whether the crowd develops collective behavioral order,we construct a simple self-propelled particle model which assumes no constantdestination nor explicit alignment interaction. We report the details of the modeland the results obtained from the numerical simulations.

22.2 Model and Simulations

Among many pedestrian models that has been previously proposed, the socialforce model [12] is the one that has been widely recognized. It assumes that eachpedestrian follows Newtonian equation of motion, which consist of the sum of self-driving force towards the destination and repulsive forces, namely the exponential“social force” from other pedestrians. In addition to the social force, anotherliterature [14] introduces normal body force and tangential friction that describesphysical contacts between people.

In order to construct a simple model and to clarify the physical meaning of crowddynamics, we start with two assumptions:

• Pedestrians do not have constant destination.• Pedestrians interact with linear elastic repulsive forces.

It may seem improbable that pedestrians have no destination on their way. However,they can in fact lose or abandon their destinations in extremely dense crowd. The

22 Collective Dynamics of Pedestrians with No Fixed Destination 245

latter point also reflects highly dense conditions where excluded volume effect playsa pivotal role in crowd dynamics.

Let us consider N polar disk particles of equal radius a moving on a two-dimensional continuous surface. The polarity of each disk is defined by an unitvector Oe. i/ D cos i Ox C sin i Oy. The equation of motion is given by

dvi

dtD ˛Oe. i/� ˇvi C

X

j

fij; (22.1)

where ri denotes the position of particle i, and the direction of their velocity is �i,i.e., vi D vi .cos �i Ox C sin �i Oy/.

Particle i drives itself with a self-propulsion force of constant magnitude ˛ alongits polarity axis while the velocity is damped by a drag force of coefficient ˇ. Thedynamics of the polarity is overdamped by a torque proportional to the angulardeviation from the direction of the velocity, as

d i

dtD � .�i � i/ ; (22.2)

where � is the damping coefficient.We assume that the interaction between particles i and j is given by a steric

repulsive force with a linear elasticity, i.e., fij D �k�2a � rij

�.ri � rj/=rij if

rij D jri � rjj < 2a and fij D 0 otherwise. Note that the momentum is conserved bythe interaction itself. Without loss of generality, we set length unit 2a D 1 and timeunit ˇ�1 D 1 and obtain rescaled equations of motion. The model is then governedby three time scales: ˛�1 is the time for a free particle to travel its own diameter,��1 is the angular relaxation time of polarity, and k�1=2 is the elastic time scale ofcollision.

The magnitude of self-propulsion force and elastic modulus are fixed as ˛ D 1

and k D 100, respectively. Under such choice of parameter values, particlespenetrate their neighbors by at most � 1% of their diameter. Therefore the elasticityis large enough to avoid unrealistic situation where particles in contact pass througheach other. Although the empirical value of elasticity of human body is not known,Helbing et al. [14] estimates it as k D 1:2 � 105 kg s�2. Given that the mass of apedestrian is 80 kg and ˇ�1 ' 0:5 s, the scaled elasticity will be k � 102, which isconsistent with the above value.

22.3 Spontaneous Ordering with Periodic BoundaryConditions

We performed molecular dynamics simulations with N D 10,000 particles on asquare plane of size L with periodic boundaries. Initial configurations are assignedrandomly in terms of particles’ position and their direction of polarity. The overlaps


between particles have been reduced by evolving the system only by the interactionforces for a sufficient time prior to each simulation. The two control parameters ofthe simulations are the angular damping coefficient � and the packing fraction

� D �Na2

4L2: (22.3)

In the region where damping is weak, the system exhibits polar ordering andclustering as shown in Fig. 22.1. To characterize the collective motion, we employas the order parameter the global polarization

� D 1

N

ˇ̌ˇ̌ˇ

NX

iD1Oe. i/

ˇ̌ˇ̌ˇ ; (22.4)

whose value is finite in the phase with polar order, and goes to zero in a globallydisordered state.For a fixed packing fraction, the growth of the order parameter is slow whendamping coefficient is small (� � 1). It is because each particle tends to keepits polarity to the same direction as given in the initial random state. As the dampingparameter increases, the speed of polarity alignment becomes faster. However,increasing the damping parameter further slows down the development of the order.Above a certain value of � , no collective motion takes place so that the systemremains disordered and isotropic. Close to this phase boundary, the time until thesystem builds up a polar order exceeds the computationally feasible time, whichmakes us difficult to identify the exact transition point. Therefore, we carriedout multiple (typically 16) runs with different initial configurations for each setof control parameters, � and � , and categorized the corresponding point in theparameter space as polar-ordered phase if � grows larger than 0.5 for at least onerealization. Obtained phase diagram is shown in Fig. 22.2.

22.4 Binary Scattering Study

In this section, we give a simple explanation to understand the mechanism thatunderlies the characteristic ordering behavior shown in previous section. Let us limitour discussion only to the binary particle collision process [15]. Here we assume thesystem is dilute enough (� ! 0) so that only the uncorrelated, binary collisions takeplace, and both the velocity and the polarity are fully relaxed before each collision.

If the damping is weak, the polarities of two particles remain unchanged, so thedirections of motion are temporally changed by the collision but eventually restoredto the original direction. Here the relative angle between the velocities does notchange before and after the collision. By contrast, if the damping is strong, thepolarities rotates themselves quickly to align to the directions of motion, so theparticles moves as if they have exchanged their momentum. Here again the absolute


Fig. 22.1 Typical snapshotsof a system in periodicboundary boxes (a) at onsetof the collective motion, and(b) at fully ordered state. Thepacking fraction � D 0:3 andthe damping parameter� D 20. The color denote thepolarity of each particle, asshown in (c)


0

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4 0.5 0.6

dam

ping

par

amet

er γ

packing fraction

disorder

order

Fig. 22.2 The �-� phase diagram obtained from molecular dynamics simulations. Gray squaresdenote states that polar order is observed and red triangles indicate the phase where the systemremains disordered. We suppose that disordered phase stretches to the upper left domain, wherewe does not have numerical results yet

value of the relative angle is maintained. For an intermediate damping parameter,the motion of two bodies align parallel due to the competing effect of the collisionand the subsequent angular damping.

Numerical results support this qualitative conjecture. Consider a binary scatteringprocess between particle i and j. Since we assume the rotational invariance, thegeometry of the moment of contact is fully specified by two scalar parameters: the

impact parameter bij Dq

r2ij � rij � .vi � vj/=vij 2 Œ0; 1�, where vij D jvi � vjjand the relative angle �ij D j�i � �jj 2 Œ0; ��, as shown in Fig. 22.3. The impactparameter shows the perpendicular offset of the two bodies’ center of mass fromhead on collision. If bij D 0 the collision is head on whereas it is a miss if bij > 1.

Instantaneous alignment of the two particles are characterized by two-particlepolarization

�.2/ D 1

2

ˇ̌Oe. i/C Oe. j/ˇ̌; (22.5)

which corresponds to the global polarization [Eq. (22.4)] with N D 2. We measurethe two-particle polarization �.2/out at an adequate time for the polarities and thevelocities to relax after the collision, and compare it to the polarization �.2/in beforethe collision. The increment,


Fig. 22.3 Schematic view of collision geometry. The geometry is defined by the relative angle �ij

and impact parameter bij. We assume that the two particles are fully relaxed in terms of velocityand polarity before each collision, therefore vi=jvij D Oe. i/

Fig. 22.4 The two-particle polarization increment ��.2/ as a function of relative angle �ij andimpact parameter bij. � is varied as denoted in the figure. For the collision geometry in the redregion, the collision makes particles align to each other (��.2/.b; �/ > 0), while in the blue regionit result in antiparallel alignment (��.2/.b; �/ < 0)

��.2/ D �.2/out � �

.2/in ; (22.6)

indicates the magnitude of parallel alignment caused by the binary scatteringprocess. Figure 22.4 depicts��.2/ as a function of the collision geometry .bij; �ij/.

Assuming that the multiparticle system is homogenous and isotropic, the prob-ability that the two particles collide in the relative angle of �ij is proportional to vij

and that impact parameters b will be equally distributed. Therefore we can obtain


-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.01 0.1 1 10 100

<Δφ

(2) >

damping parameter

Fig. 22.5 Average alignment tendency integrated over all collision geometries, as a function ofthe damping parameter � . Note that for large values of � , the tendency drops to negative values

average tendency to align parallel for a specific value of � by estimating the expectedvalue with an integration weight of “scattering cross section”

h��.2/i D 1

C

Zdb

Zd�ij

ˇ̌ˇ̌sin

��ij

2

�ˇ̌ˇ̌ ��.2/.bij; �ij/; (22.7)

where C is a normalization constant.The result shown in Fig. 22.5 indicates that the alignment tendency hits its

peak at � � 1. For � ! 0, which corresponds to the regime where angularrelaxation is slow, h��.2/i goes to zero. For large � , namely � ! 1, h��.2/i hasa negative value. This picture have two consistency with the result obtained fromthe multiparticle simulations; (1) The ordering in many-body system is the fastestin the parameter region that maximize the value of h��.2/i; (2) The transition in thedilute system occurs at � � 10, where h��.2/i changes its sign. These points implythat at least for the dilute limit � ! 0, the onset of collective motion arise from therepetition of binary collision process.

22.5 Flow in a Pipe

In order to validate the correspondence of the model with actual pedestrianphenomena, we performed multiparticle simulations in a “pipe”, i.e., a rectangulararea with periodic boundaries in the longitudinal direction and fixed repulsive


Fig. 22.6 Snapshots of the simulations carried out under “pipe” condition. The system consist ofN D 3200 particles and is periodic for x-direction while bounded by elastic slipping walls fory-direction. The width of the pipe is 20:0, � D 0:5; � D 0:01. At t D 300, three-lane structure isobserved. (a) snapshot at t D 0, (b) t D 100, (c) t D 300, (d) t D 500

boundaries in the lateral direction. The interaction between the particles and therepulsive boundaries is assumed to be similar to the particle-particle interaction,that is, the interaction potential is elastic and frictionless. Starting from randominitial condition, the system develops into two lanes of particle flow moving inopposite directions for certain sets of parameters (Fig. 22.6). Three-lane structure isalso observed in a transient state. These results indicate that the model can reproducethe lane formation, which is one of the basic self-organization phenomena observedin pedestrian crowd [12]. The stability of lane structures and their dependency to thewidth of the pipe are subject of future investigation.

22.6 Summary and Discussion

In order to close the gap between theory on self-propelled particles and pedestriandynamics study, we proposed a self-propelled particle model with repulsive interac-tion, and examined its collective behavior through many-body simulations. Binaryscattering studies demonstrate the microscopic mechanism underlying the transitionfrom a disordered to a polar-ordered phase.

Our model assumes that the collision process itself is elastic, i.e., no dissipation istaken into account. Nevertheless, the effective inelasticity introduced by the angulardamping allows the formation of clusters, or “flocks,” similar to that of granulargases. Of course, this argument is not exact because the many-body correlationcannot be ignored once local clusters are formed. Still, it provides a qualitative and,


to some extent, quantitative explanation. We look forward to a further discussion onmany-body effect and on cluster-cluster interaction.

The conventional social force model [12] assumes that each pedestrian experi-ences the social force from other pedestrians in his/her eyesight, which is describedas an exponentially decaying repulsion. We expect that any isotropic, short-ranged repulsive potential, including exponential one, would not deviate the overallproperty of the system from the results we obtained with linear elastic repulsion.On the other hand, introduction of anisotropic potential that reflects the fact thatpedestrians react stronger to the situation in front of them is not clear and yet to bediscovered.

It is known that high crowd density leads to a turbulent movement of pedestriansand increases the risk of crowd disaster [16]. In spite of social demands to preventsuch accidents from occurring and from spreading, their mechanism is yet to beuncovered, since experiments cannot be carried out due to ethical reasons, andobservational data are hardly available. Previous pedestrian models assume thatevery agent is aware of its own destination and keeps driving itself until it reaches tothat point. However, there are circumstances when pedestrians are not so consciousof where they are heading to. In fact we scanned the footage from the crowd disasterhappened in Germany in 2010 and found that people sometimes behave as if theyhave lost or abandoned their initial destination in extremely dense crowd.

To this end, we verified that our model, which has no fixed destination, coulddisplay bidirectional lanes similar to what is observed in pedestrian flows in astraight pathway. Here, the damping parameter � in the model can be regarded as thequickness of one’s reaction to a contact with neighbor walkers. However, the phasediagram shows that in higher density, the order develops for a broader range of theparameter, which does not meet with the empirical facts. By improving the modelwe expect that our result can lead to a further understanding on the mechanism ofcrowd disasters.

Open Access This book is distributed under the terms of the Creative Commons Attribution Non-commercial License which permits any noncommercial use, distribution, and reproduction in anymedium, provided the original author(s) and source are credited.

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10. Burstedde C, Klauck K, Schadschneider A, Zittartz J (2001) Physica A 295:507


11. Kirchner A, Nishinari K, Schadschneider A (2003) Phys Rev E 67:05612212. Helbing D, Molnár P (1995) Phys Rev E 51:428213. Yu WJ, Chen R, Dong LY, Dai SQ (2005) Phys Rev E 72:02611214. Helbing D, Farkas I, Vicsek T (2000) Nature 407:48715. Hanke T, Weber CA, Frey E (2013) Phys Rev E 88:05230916. Helbing D, Johansson A, Al-Abideen H (2007) Phys Rev E 75:046109

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